Theorem opid 3638

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Hypothesis

Ref	Expression
opid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
opid	⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 3458	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2092	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 3521	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		opid.1	. . 3 ⊢ 𝐴 ∈ V
5	4, 4	dfop 3619	. 2 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6		dfsn2 3458	. 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	3, 5, 6	3eqtr4i 2118	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1289   ∈ wcel 1438  Vcvv 2619  {csn 3444  {cpr 3445  ⟨cop 3447

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453

This theorem is referenced by:  dmsnsnsng  4903  funopg  5042